Who's Afraid of Nagelian Reduction? F. Dizadji-Bahmani, R. Frigg and S. Hartmann February 2009 Abstract We reconsider the Nagelian theory of reduction and argue that, con- trary to a widely held view, it is the right analysis of inter-theoretic reduction. For one, its purported successor, so-called new wave reduc- tionism, turns out collapses into a sophisticated version of Nagelian reduction and hence does not provide an alternative. For another, the alleged difficulties of the Nagelian theory either vanish upon closer in- spection, or turn out to be interesting philosophical questions rather than knock-down arguments. Contents 1 Introduction 2 2 Statistical Mechanics - A Reductionist Enterprise 3 3 Nagelian Reduction 8 3.1 The Nagel-Schaffner Model . 8 3.2 Problems for the Nagel-Schaffner Model . 13 4 New Wave Reductionism 15 5 Nagelian Reduction Reconsidered 19 6 Conclusion 24 1 Bibliography 24 1 Introduction The purpose of this paper is to examine synchronic inter-theoretic reduction, i.e. the reductive relation between pairs of theories each of which describes the same phenomena and which are simultane- ously valid to various extents.1 Examples of putative synchronic inter-theoretic reductions are macro economics to micro economics, chemistry to atomic physics, and thermodynamics (TD) to statistical mechanics (SM). The latter will be our test case: what is it to say that TD reduces to SM? The central contention of this paper is that Nagel's account of reduc- tion essentially gives the right answer to this question. We first turn our attention to the Nagelian model of reduction and consider some of the problems that it allegedly faces. We then discuss its successor, so- called new wave reductionism (NWR), which is nowadays commonly advocated in its place. Our conclusion is twofold. First, we argue that upon closer inspection NWR collapses into a sophisticated version of Nagelian reduction, which, for reasons that will become clear as we proceed, we refer to as Nagel-Schaffner Reduction (NSR). Hence, re- ceived wisdom notwithstanding, NWR does not provide an alternative to NSR. Second, we reconsider the alleged difficulties of NSR and con- clude that not only are they far from being as insurmountable as it they are often said to be; in fact most of them vanish upon closer inspection and those that don't turn out to be interesting philosoph- ical issues rather then knock-down arguments. So NSR is alive and well and can be used as a regulative model for reductionist research programmes. 1There are, of course, various other types of reductive relations, most notably diachronic theory reductions an example of which is Newtonian and relativistic mechanics. See Nickles (1975). For an in-depth discussion of such cases see Batterman (2002). 2 2 Statistical Mechanics - A Reduction- ist Enterprise SM is the study of the connection between micro-physics and macro- physics. TD correctly accounts for a broad range of phenomena we ob- serve in macroscopic systems like gases and solids. It does so by char- acterizing the behavior of such systems as governed by laws which are formulated in terms of macroscopic properties such as volume, pres- sure, temperature and entropy. The aim of statistical mechanics is to account for this behaviour in terms of the dynamical laws governing the microscopic constituents of macroscopic systems and probabilistic assumptions. There is a broad consensus, among physicists and philosophers alike, that SM is a reductionist enterprise. The following quotes are indica- tive of this: `We know today that the actual basis for the equivalence of heat and dynamical energy is to be sought in the ki- netic interpretation, which reduces all thermal phenomena to the disordered motions of atoms and molecules' (Fermi 1936 p.ix). `The explanation of the complete science of thermodynam- ics in terms of the more abstract science of statistical me- chanics is one of the greatest achievements of physics.' (Tol- man 1938, 9) `The classical kinetic theory of gases is [a] case in which thermodynamics can derived nearly from first principles.' (Huang 1963, Preface) Further statements pulling in the same direction can be found in Dougherty (1993, 843), Ehrenfest & Ehrenfest (1912, 1), Goldstein (2001, 40), Khinchin (1949, 7), Lebowitz (1999, 346), Ridderbos (2002, 66), Sklar (1993, 3) and Uffink (2007, 923). What is meant by reduction? That practitioners of SM do not really discuss the issue is not really a surprise; however, it should rise some 3 eyebrows that by and large philosophers working on the foundations of SM also only rarely address this issue. So the pressing question remains: what notion of reduction is at work in the context of TD and SM? Different statements of the reductive aims of SM emphasise different aspects of reduction (ontological, explanatory, methodological, etc.), but all agree that a successful reduction of TD to microphysics in- volves the derivation of the laws of TD from the laws of microphysics plus probabilistic assumptions. This has a familiar ring to it: deduc- ing the laws of one theory from another, more fundamental one, is precisely what Nagel (1961) considers a reduction to be. Indeed, the Nagelian model of reduction seems to be the (usually unquestioned and unacknowledged) `background philosophy' of SM. One could lay the case to rest at this point if Nagel's model of reduc- tion was generally accepted as a viable theory of reduction. However, the contrary is the case. As is well known, the Nagelian model of reduction was from its inception widely criticised, and is now gen- erally regarded as outdated and misconceived. Representative for a widely shared sentiment about Nagel's account is Primas, who notes that `there exists not a single physically well-founded and nontrivial example for theory reduction in the sense of Nagel...' (1998, 83). This leaves us in an awkward situation. On the one hand, if Nagel's account really is the philosophical backbone of SM, then we have an (allegedly) outdated and discarded philosophy at work in what is gen- erally accepted as the third pillar of modern physics alongside rela- tivity and quantum theory! This is unacceptable. If we want to stick with Nagelian reduction the criticisms have to be rebutted. On the other hand, if, first appearances notwithstanding, Nagel's account is not the philosophical backbone of SM, what then is? In other words, the question we then face is: what notion of reduction, if not Nagel's, is at work in SM? This dilemma is not recognised in the literature on SM, much less seriously discussed. But when raised in informal discussion { at re- ceptions and in the corridors of conference hotels { one is usually told to embrace the second option: Nagelian reduction is outdated and 4 discarded but the so-called `New Wave Reductionism' associated with the work of Churchland and Hooker provides a model of reduction that avoids the pitfalls of Nagelian reduction while providing a viable philosophical backbone of SM. In what follows we argue that this is an empty promise. Before delving into the discussion of different accounts of reduction, let us introduce two cases against which we test our claims: the Ideal Gas Law and the Second Law of thermodynamics. These are gener- ally considered to be paradigm cases of reduction and hence serve as a benchmark for accounts of reduction. Ideal Gas Law. The state of a gas is specified by three quantities: pres- sure p, volume V , and temperature T . A gas is ideal if it consists of particles without spatial extension (point particles) which do not inter- act with each other. Needless to say, there are no ideal gases in nature, but as long as the pressure is low, real gases can be treated as ideal gases to a very good approximation (since the volume of molecules is extremely small compared to the volume occupied by the gas and the inter-molecular forces are negligable). If such a gas is in equilibrium (i.e. if it is evenly distributed over V , and p and T do not change over time), volume and temperature are related to one another by the so-called Ideal Gas Law: p V = k T , where k is a constant. Let us call this law together with the qualifications about its scope the thermal theory of the ideal gas. Consider a gas consisting of n particles of mass m confined to a vol- ume V , for instance a vessel on the laboratory table. Each particle has a particular velocity ~v, and its motion is governed by Newton's equations of motion. Assume, furthermore, that we are given a veloc- ity distribution f(~v), specifying what portion of all particles move in direction ~v. The exact form of this distribution is immaterial at the moment. Let us call Newtonian mechanics plus the assumptions just mentioned the kinetic theory of the ideal gas. The aim now is to derive the law of the thermal theory of the ideal gas from the laws kinetic theory.2 2For details see Greiner et al (1993, 12-15) or Pauli (1973, 94-103). 5 Pressure is defined (in Newtonian physics) as force per surface: p = FA =A, where A is surface (for instance a section of the kitchen table) and FA the force acting perpendicular on the surface (for instance the gravitational force exerted on the table by a glass placed on it). If a particle crashes into the wall of the vessel and is reflected it exerts a force onto the wall, and the exact magnitude of this force follows immediately from Newton's equation of motion. We now assume that all particles in the gas are non-interacting and perfectly elastic point particles.